Uniqueness property for quasiharmonic functions

In this paper we consider class of continuous functions, called quasiaharmonic functions, admitting best approximations by harmonic polynomials. In this class we prove a uniqueness theorem by analogy with the analytic functions

Nonlinear self-adjointness and conservation laws

The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the strict self-adjointness and quasi self-adjointness introduced earlier by the author. It is shown that the equations possessing the nonlinear self-adjointness can be written equivalently in a strictly self-adjoint form by using appropriate multipliers. All linear equations possess the property of nonlinear self-adjointness, and hence can be rewritten in a nonlinear strictly self-adjoint. For example, the heat equation $u_t - \Delta u = 0$ becomes strictly self-adjoint after multiplying by $u^{-1}.$ Conservation laws associated with symmetries can be constructed for all differential equations and systems having the property of nonlinear self-adjointness

A Group Theoretical Identification of Integrable Equations in the Li\'enard Type Equation x¨+f(x)x˙+g(x)=0\ddot{x}+f(x)\dot{x}+g(x) = 0 : Part II: Equations having Maximal Lie Point Symmetries

In this second of the set of two papers on Lie symmetry analysis of a class of Li\'enard type equation of the form $\ddot {x} + f(x)\dot {x} + g(x)= 0$, where over dot denotes differentiation with respect to time and $f(x)$ and $g(x)$ are smooth functions of their variables, we isolate the equations which possess maximal Lie point symmetries. It is well known that any second order nonlinear ordinary differential equation which admits eight parameter Lie point symmetries is linearizable to free particle equation through point transformation. As a consequence all the identified equations turn out to be linearizable. We also show that one can get maximal Lie point symmetries for the above Li\'enard equation only when $f_{xx} =0$ (subscript denotes differentiation). In addition, we discuss the linearising transformations and solutions for all the nonlinear equations identified in this paper.Comment: Accepted for publication in Journal of Mathematical Physic

Public procurement’s legal regulation through the medium of competitive procurement in the Republic of Kazakhstan

The aim of the article is to study the problems of legal regulation of government procurement in Kazakhstan from the point of view of the latest domestic and foreign researches. The methodological basis of the study was the legislative acts, statutory documents on the theme of work, fundamental theoretical works, results of practical studies of prominent domestic and foreign scientists. According to the results, it is important to note that the closed competitive tender is characterized by the fact that the offer to participate in it, is drawn to a certain circle of persons at the discretion of the competition’s organizer. The authors of the study came to the main conclusion that the law on state procurement is being improved, the Law "On State Procurement" is being updated, and orders are issued on topical issues, namely on combating corruption.peer-reviewe

Ordinary differential equations which linearize on differentiation

In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.Comment: 9 page

On the hierarchy of partially invariant submodels of differential equations

It is noticed, that partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduce a hierarchic structure in the set of all PISs of a given system of differential equations. By using this structure one can significantly decrease an amount of calculations required in enumeration of all PISs for a given system of partially differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. In this framework the complete classification of regular partially invariant solutions of ideal MHD equations is given

Titanomagnetites with Magnetite–Ulvospinel breakdown structures. Coercive properties: modeling and experiment

© 2015, Pleiades Publishing, Ltd. One-, two-, and three-dimensional models of magnetite–ulvospinel exsolution structures are constructed based on the experimental data. The models describe the zero state and the state of remanent saturation magnetization. A distinctive feature of these models is the calculation of the magnetostatic interaction between the ferromagnetic matrices of magnetite separated by paramagnetic ulvospinel lamella. The critical sizes of matrix transition from a single-domain to two-domain state are calculated. Based on the calculations, it is concluded that the matrices can only have a singleor two-domain structure. The ratios Mrs/Ms for the matrices of various sizes and the dependences describing the growth in the induced magnetic moment Mi in the fields of up to 25 mT are calculated. The sizes of the exsolution structures in two samples are estimated by electron microscopy. The values observed in the experiments closely agree with the predictions by the suggested models

Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations

The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations implies the linearizability of systems of partial differential equations corresponding to those complex ordinary differential equations. The invertible complex transformations can be used to obtain invertible real transformations that map a system of nonlinear partial differential equations into a system of linear partial differential equation. Explicit invariant criteria are given that provide procedures for writing down the solutions of the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single research paper "Linearizability criteria for systems of two second-order differential equations by complex methods" which has been published in Nonlinear Dynamics. Due to citations of both parts I and II these are not replaced with the above published articl